"I think it's sad that in the media they're just saying that a German has died, and that's why all the neo-Nazis and hooligans are out, but the media should describe who died, and what skin colour he had, because I don't think they'd be doing all this if they knew."

State and local officials appealed for calm as thousands of people took to the streets in Monday's protests that turned violent after injuries caused by fireworks thrown from both sides.

Local media reported Neo-Nazis performing the Nazi salute, and local TV showed footage of skinheads chasing a man.

"The scenes of people going after those who look like foreigners scare us. We want to show that Chemnitz has another side that is cosmopolitan and opposes xenophobia," Tim Detzner, head of the radical left party in Chemnitz, said at the anti-fascist rally.

Demonstrators waving German and Bavarian flags were also present nearby, some breaking through police barriers aimed at keeping the two sides apart.

Spokesperson for German Chancellor Angela Merkel's said Germany would not tolerate "vigilante justice".

"We don't tolerate such unlawful assemblies and the hounding of people who look different or have different origins, and attempts to spread hatred on the streets," Steffen Seibert said at a regular news briefing.

The far-right has surged across Europe during the continent's migrant crisis and increasing political rhetoric against Islam.

Germany's far-right Alternative for Germany party (AfD) is now the main opposition party, and Merkel faces opposition within her governing coalition for her immigration policies.

More than a million people fleeing war and poverty in the Middle East have sought refuge in Germany since 2015.